January 18, 1917] 
NATURE 
387 
HISTORY OF MATHEMATICS. 
Historical Introduction to Mathematical Litera- 
ture. By Prof. G. A. Miller. Pp. xiii+ 302. 
(New York: The Macmillan Co.; Lon- 
-don: Macmillan and Co., Ltd., 1916.) Price 
7s. net. 
ges of the bistong of mathematics are 
better left to specialists, who still have 
-plenty of occupation in clearing up doubtful 
“points and amending errors. But there is a grow- 
ing opinion among teachers that not only for 
themselves, but also for their pupils, some know- 
ledge of the course of mathematical discovery is 
eminently desirable. Besides being a factor in a 
-general education, it is stimulating to the 
learner, and supplies to the teacher a view of 
human activity and invention which ought to be 
“suggestive from the psychological side. If there 
“be a ‘‘natural” order of learning mathematics, it 
‘cannot be wholly different in the race and the in- 
dividual; though, of course, this consideration 
ought not to be turned into a fad. A year should 
mot be wasted on heuristic acquisition of the 
multiplication table. 
To serve the purposes indicated, we want books 
~which are not too long, put the main facts into 
proper perspective, avoid doubtful assertions, 
sand show the trend of mathematics at the present 
‘time. In all these respects Prof. Miller seems to 
wus to be successful. As to the perspective, a con- 
siderable proportion of the space is given to 
‘modern mathematics; this is quite justified by the 
remarkable progress, and in some ways revolu- 
‘tion, of recent times. But the earlier history is 
by no means neglected; thus we have accounts of 
ancient and medieval arithmetic, geometry, and 
algebra, including the theory of irrationals*—all in 
broad outline, but very well arranged. Among 
modern ‘topics, we have a chapter on the de- 
‘velopment of mathematics since the close’ of the 
‘eighteenth century, and one on mathematical 
literature; the last ought to be very useful to 
those who are serious students of the subject. 
The last chapter gives brief biographies of 
twenty-five deceased mathematicians, ranging 
from Euclid and Archimedes to Lie and Poincaré. 
‘The list could scarcely be improved upon, and 
the notices, on the whole, are excellent. For in- 
stance, justice is done to Cauchy’s great achieve- 
ments, at least those in pure mathematics, and the 
author searcely professes to deal with applied 
mathematics. At the same time, notice is taken 
of Newton’s theory of gravitation and of Poin- 
‘caré’s work on celestial mechanics, so that we 
«cannot help being surprised when we find nothing 
said about Rowan Hamilton’s contributions to 
dynamics or even his researches on systems of 
rays. It is curious how many seem to think of 
Hamilton as the inventor of quaternions and 
of nothing else. 
The appendix gives a brief list of books, and is, 
we-think, the most uneven part of the work; it 
almost seems as if the author had looked round 
his bookshelves and put down the titles of those 
NO. 2464, VOL. 98] 
volumes that caught his eye. For instance, under 
“Bibliographies and Encyclopedias” we have, 
among twenty entries, Mr. Somerville’s biblio- 
graphy of non-Euclidean geometry; the value of 
this is indisputable, but it is far too special a - 
work for a list of this kind. Again, under “ Teach- 
ing and Philosophy,” we have the “ Monographs ” 
edited by Prof. J. W. A. Young; these are quite 
special things, like the Cambridge Tracts and 
other such publications, and to put them here 
among eighteen entries shows a lack of propor- 
tion. 
Two things may strike the reader of the bio- 
graphies: the full names are not always stated, 
and no indication is given of Jewish nationality. 
The last is a small matter; but the comparatively 
large number of Jews who have become eminent 
mathematicians and physicists is certainly remark- 
able. 
Prof. Miller has the great merits of being lively 
and enthusiastic, and appreciating the beauties of 
his science. His anecdotes and obiter dicta are 
always interesting, and sometimes highly amus- 
ing; for instance, Abel writes of Cauchy: “Ses 
travaux sont excellents, mais il écrit d’une maniére 
trés confuse.’ ‘Unless we are greatly mistaken, 
Abel deserves this criticism much more than 
Cauchy. Again, it will be news to most people 
that “Omar Alkhayami” (FitzGerald’s Omar 
Khayyam) ‘‘made. a statement in his algebra 
which seems to imply that he was able to deter- 
mine the coefficients of the successive terms in the 
expansion of a binomial raised to any positive 
integral power.” 
We hope that copies of this book will find their 
way into many of our school libraries; quite a 
large part of it ought to be thoroughly enjoyed 
by a mathematical boy. It is well printed, too, and 
comparatively cheap. Gi5B aM: 
OUR BOOKSHELF. 
The Origin of the Earth. By Thomas Chrowder 
Chamberlin. Pp. xi+271. (Chicago: The 
University of Chicago Press; London: At the 
Cambridge University Press, 1916.) Price 6s. 
net. 
Tuts book forms the third of a series of publica- 
tions intended to “present the complete results of 
series of investigations which have previously 
appeared only in scattered articles, if published 
at all.” Needless to say, it is occupied mainly 
with a presentation of the planetesimal hypo- 
thesis, associated with the name of the author 
and his collaborator, Prof. F. R. Moulton. The 
original investigations on the planetesimal theory 
have perhaps been rather more scattered than 
most, so that an account of them in a compact 
and continuous form is especially welcome. 
Prof. Chamberlin’s theory is frankly tentative 
and speculative, and the reader is invited through- 
out to form his own judgment of the value of 
what is offered for his acceptance. The reader will 
proceed with caution, as indeed he is advised to 
do, for the progress of astronomy makes it evident 
